In experiments to measure photon flux for example, where the measured signal is the total count of photons recorded in a fixed time interval, successive measurements will produce different results due to statistical variation. Even if the flux is constant, successive values will be distributed according to a Poisson probability distribution with mean and variance both equal to the average expected count for that time interval. This fluctuation makes it difficult to detect small changes in flux. However, if the measurement time is extended, more counts are recorded in the time interval and the relative statistical variance is reduced so that smaller relative changes in flux can be detected. This principle applies to any experiment where the recorded counts obey Poisson counting statistics so that for example electrons, ions or other fundamental particles may be counted rather than photons.
FIG. 1 shows a practical example where a single measurement involves counting the total number of x-ray photons with energy from 0.1 to 10 keV that hit an x-ray detector within a fixed time while an electron beam is held at a fixed position on the specimen. Measurements are taken on a 2-D grid (256 by 192) of pixel positions and the results plotted as a grey scale image or “x-ray map”, with an average of four counts per pixel. In FIG. 1, the statistical variation in pixel counts produces a random “noisy” appearance to the image that obscures any detail.
Although the total count does not show any clear structure in the image, the compositional variations in the specimen are likely to cause changes in the distribution of counts at different energies at each pixel. In an energy dispersive spectrometer (EDS) the x-ray spectrum is typically recorded as a histogram showing the number of photons recorded in each of a number of contiguous energy intervals called “channels” across the full range of energies. For a so-called “spectrum image” spectra are recorded at every position on a grid of pixel positions.
U.S. Pat. No. 6,584,413 describes how such spectrum images can be analysed by multivariate statistical techniques to identify systematic variations in spectra and discover properties of the sample. For this approach, the relative magnitude of systematic and random fluctuations is taken into account by some weighting scheme. To do this, the assumption is usually made that the random nature of Poisson distributed count values can be approximated by a distribution with standard deviation equal to the square root of the count value. While this is a satisfactory approximation for large count values, the Poisson distribution becomes severely asymmetric when the average count value falls below 1. As a consequence, multivariate statistical methods become less effective at very low count values where the statistical noise does not follow a normal probability distribution.
When the recording time is short or the counting rate is low, there may be very few counts in a single pixel spectrum. For example, in FIG. 1, the average single pixel spectrum contains only 4 counts for the total energy range 0.1 to 10 keV, so plotting the energy histogram for a single pixel spectrum is not particularly helpful. Therefore, it is common for spectra to be summed over all pixels and the sum spectrum over the whole field viewed on a graphical display to show an aggregate response for the specimen. As shown in FIG. 2, the sum spectrum reveals a number of peaks characteristic of the elements present in the specimen. In an inhomogeneous sample, some elements will be present at every position and some elements will only be present in discrete areas within the field of view.
If an energy region of interest “ROI”, is defined spanning a particular peak, then the integral count in this ROI can be calculated from the histogram for a single pixel and this count can be plotted as a function of pixel position to produce an element x-ray map. In FIG. 2 such maps have been created for oxygen (O) and aluminium (Al). Like the total count image in FIG. 1, the oxygen map in FIG. 2 does not show any structure, but the aluminium map does show some spatial variation. This spatial variation indicates that there are regions of different elemental composition in the sample.
In general it is not clear which ROI will produce x-ray maps that demonstrate variation in composition. Furthermore, the statistical variation in counts in the sum spectrum may obscure small peaks for elements that only occur in small areas in the field of view. For example, if the image consists of 256×256 pixels, then a small area of 4×4 pixels contributes a fraction of only 2.4E-04 to the total sum spectrum. Thus, if a 4×4 pixel region exhibits a different x-ray spectrum from most of the field of view, then the influence on the overall sum spectrum may be smaller than the statistical scatter in the sum spectrum.
One known way to detect rare pixels where the spectrum contains peaks not present in most of the other pixels is to compute the “maximum-pixel spectrum” (“MPS”) (“Maximum pixel spectrum: a new tool for detecting and recovering rare, unanticipated features from spectrum image data cubes”, D. S. Bright & D. E. Newbury, Journal of Microscopy, Vol. 216, Pt 2, November 2004, pp. 186-193). For every pixel in the grid, the count recorded in a particular energy channel is noted and the maximum count over all pixels is taken as the value for the maximum pixel spectrum at this energy channel. Bright & Newbury disclose a sum spectrum for a grid of 256×200 pixels taken with 0.5 s dwell per pixel and approximately 4000 cps count rate per pixel so that the average spectrum for a single pixel contains 2000 counts. They also show the corresponding maximum-pixel spectrum, which exhibits additional peaks for O, Cu, Si, Cl, K, Cr that are not obvious in the sum spectrum and correspond to small areas in the field with a high concentration of these elements. X-ray maps constructed from the ROI around the peaks for these elements would show where there are pixels with a high concentration of these elements. The example maximum-pixel spectrum (MPS) discussed by Bright & Newbury looks like a recognisable x-ray spectrum and represents count values for single pixels while the sum spectrum represents the total of all pixels so the magnitudes are very different.
However, Bright & Newbury show that MPS and sum spectra can be usefully compared by scaling to the maximum peak. This scaling and comparison works well for such high-count spectra where the average count per pixel varies between about 1 and 100 in a single energy channel. However, the Bright & Newbury data took 9 hours to record and in practice much smaller acquisition times may be used so that the average count per pixel may be much less than 1. In such lower count conditions the MPS exhibits unusual characteristics.
FIG. 3a shows the results of an experiment where data have been recorded for a total of one minute on a grid of 256×192 pixels at an average count rate of 2500 counts per second (cps). The spectra contain 2048 channels, each of 5 eV width. In this case the MPS and SUM spectra have been scaled to the Si K peak at 1.74 keV. The mean count for the Si K peak channel is only 0.04 counts per pixel and the maximum over all pixels is 3 counts. The scaled MPS shows discrete levels corresponding to 1, 2 and 3 counts and does not look like a typical x-ray spectrum. In this example, typical x-ray peaks are much broader than the 5 eV channel width and, as Bright & Newbury suggest, each channel count can be replaced with the total of several channels symmetrically disposed about the channel energy. This reduces statistical scatter at the expense of energy resolution and the total over this symmetric set of channels is used instead the channel value when computing the MPS for each energy channel. In FIG. 3b, a total for 15 channels covering an energy region of 75 eV in width is used for every channel in a pixel spectrum. In this case, the mean for the Si K peak is now 0.56 counts over all pixels and the maximum over all pixels is 7 counts. Although the scaled MPS now demonstrates characteristic peaks at 1.5 and 1.74 keV, in the rest of the MPS it is not obvious what structures represent real elemental peaks and what is just statistical noise.
A problem with MPS occurs because when the underlying average count per pixel is much less than 1, the maximum can only take values of 0, 1, 2 etc and therefore can be orders of magnitude greater than the mean. In contrast, when the average count per pixel is much greater than 1, the maximum will only be slightly greater than the mean. Therefore, in general, there is no single scale factor that will allow the MPS to be compared with the sum spectrum in order to detect differences from average behaviour over the field of view. Furthermore, with low count data, the MPS is so noisy that it is not always possible to decide whether an elemental peak is present or absent.
Thus there are a number of known techniques for producing spectra in order to visually allow the determination of structure or compositional information relating to the object from which the spectrum is obtained. Some of these techniques have been combined so as to provide additional information to a user. However, despite this, there remains a need to provide an effective method which can be used to indicate which parts of a spectrum actually relate to underlying structure within the data.